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Hilbert Transform

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The Hilbert transform (and its inverse) are the integral transform

g(y)=H[f(x)]=1/piPVint_(-infty)^infty(f(x)dx)/(x-y)
(1)
f(x)=H^(-1)[g(y)]=-1/piPVint_(-infty)^infty(g(y)dy)/(y-x),
(2)

where the Cauchy principal value is taken in each of the integrals. The Hilbert transform is an improper integral.

They will be implemented in a future version of the Wolfram Language as HilbertTransform[f, x, y] and InverseHilbertTransform[g, y, x], respectively.

In the following table, Pi(x) is the rectangle function, sinc(x) is the sinc function, delta(x) is the delta function, AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x) and AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x) are impulse symbols, and _1F_1(a;b;x) is a confluent hypergeometric function of the first kind.

f(x)g(y)
sinxcosy
cosx-siny
(sinx)/x(cosy-1)/y
Pi(x)1/piln|(y-1/2)/(y+1/2)|
1/(1+x^2)-y/(1+y^2)
sinc^'(x)(1-cosy-ysiny)/(y^2)
delta(x)-1/(piy)
AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x)y/(pi(1/4-y^2))
AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x)-1/(2pi(1/4-y^2))
e^(-x^2)-e^(-y^2)erfi(y)

See also

Abel Transform, Fourier Transform, Improper Integral, Integral Transform, Titchmarsh Theorem, Inverse Hilbert Transform, Wiener-Lee Transform

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References

Bracewell, R. "The Hilbert Transform." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 267-272, 1999.Papoulis, A. "Hilbert Transforms." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 198-201, 1962.

Referenced on Wolfram|Alpha

Hilbert Transform

Cite this as:

Weisstein, Eric W. "Hilbert Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertTransform.html

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